Exactly-Solvable Self-Trapping Lattice Walks. Part I: Trapping in Ladder Graphs

TL;DR

This paper derives exact probabilities and mean trapping times for growing self-avoiding walks on ladder graphs, providing insights into trapping behavior in simplified lattice geometries.

Contribution

It introduces generating functions for trapping probabilities on ladder graphs and calculates exact mean trapping times, advancing understanding of self-avoiding walk trapping.

Findings

Mean trapping time on square ladder: 17 steps

Mean trapping time on triangular ladder: approximately 19.6 steps

Provides exact probability distributions for trapping in simplified lattice geometries

Abstract

A growing self-avoiding walk (GSAW) is a stochastic process that starts from the origin on a lattice and grows by occupying an unoccupied adjacent lattice site at random. A sufficiently long GSAW will reach a state in which all adjacent sites are already occupied by the walk and become trapped, terminating the process. It is known empirically from simulations that on a square lattice, this occurs after a mean of 71 steps. In Part I of a two-part series of manuscripts, we consider simplified lattice geometries only two sites high ("ladders") and derive generating functions for the probability distribution of GSAW trapping. We prove that a self-trapping walk on a square ladder will become trapped after a mean of 17 steps, while on a triangular ladder trapping will occur after a mean of 941/48 (~19.6 steps). We discuss additional implications of our results for understanding trapping in the "infinite" GSAW.